Consider a parabola `y=Ax^(2)+B, -x_(0) le x le x_(0)`.

If this curve is rotated about y axis, we get a paraboloid surface. The volume below this surface & above x–z plane is given by `V=(piAx_(0)^(4))/2+piBx_(0)^(2)=("volume of cylinder ABCD")/2+"Volume of cylinder CDEF"`
Use the above result to answer following question.
A cyllindrical container of height ‘h’ and radius ‘a’ is two-third filled. Find maximum angular velocity at which liquid can be rotated without spilling it.

Options

A. a. `sqrt((4gh)/(3a^(2)))`
B. b. `sqrt((2gh)/(3a^(2)))`
C. c. `sqrt((gh)/(a^(2)))`
D. d. `sqrt((gh)/(2a^(2)))`

Explanation

Profile of rotating liquids is given by

`y=(omega^(2)x^(2))/(2g)`
Putting `x=1,y=(omega^(2)a^(2))/(2g)`
Volume of liquid in fig (b) is written as `=(pia^(2)xx(omega^(2)a^(2))/(2g))/2+pia^(2)(h-(omega^(2)a^(2))/(2g))`
Equating to volume in figure (a), we get
`pia^(2) (2h)/3 =(pia^(2)omega^(2)a^(2))/(4g)+pia^(2)[h-(omega^(2)a^(2))/(2g)]rArr omega=sqrt((4gh)/(3a^(2)))`